LMFDB - Embedded newform 75.22.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(1,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$209.608008215$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{649}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 162 $ x^2 - x - 162
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-12.2377$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1271.96 q^{2} -59049.0 q^{3} -479282. q^{4} -7.51077e7 q^{6} -6.32076e8 q^{7} -3.27711e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q+1271.96 q^{2} -59049.0 q^{3} -479282. q^{4} -7.51077e7 q^{6} -6.32076e8 q^{7} -3.27711e9 q^{8} +3.48678e9 q^{9} +5.97585e10 q^{11} +2.83011e10 q^{12} -7.38499e11 q^{13} -8.03972e11 q^{14} -3.16321e12 q^{16} +8.35876e12 q^{17} +4.43503e12 q^{18} +4.19061e13 q^{19} +3.73234e13 q^{21} +7.60101e13 q^{22} -4.48926e13 q^{23} +1.93510e14 q^{24} -9.39338e14 q^{26} -2.05891e14 q^{27} +3.02943e14 q^{28} -2.76669e15 q^{29} +8.36452e15 q^{31} +2.84914e15 q^{32} -3.52868e15 q^{33} +1.06320e16 q^{34} -1.67115e15 q^{36} +1.77675e16 q^{37} +5.33027e16 q^{38} +4.36077e16 q^{39} +1.45253e17 q^{41} +4.74737e16 q^{42} -1.24744e17 q^{43} -2.86412e16 q^{44} -5.71014e16 q^{46} -4.28566e17 q^{47} +1.86784e17 q^{48} -1.59026e17 q^{49} -4.93577e17 q^{51} +3.53950e17 q^{52} +4.77017e17 q^{53} -2.61884e17 q^{54} +2.07138e18 q^{56} -2.47452e18 q^{57} -3.51911e18 q^{58} +1.61959e18 q^{59} -3.76882e18 q^{61} +1.06393e19 q^{62} -2.20391e18 q^{63} +1.02577e19 q^{64} -4.48832e18 q^{66} +2.81797e18 q^{67} -4.00621e18 q^{68} +2.65086e18 q^{69} +1.00228e19 q^{71} -1.14266e19 q^{72} +1.72739e19 q^{73} +2.25995e19 q^{74} -2.00849e19 q^{76} -3.77719e19 q^{77} +5.54670e19 q^{78} -3.28276e19 q^{79} +1.21577e19 q^{81} +1.84755e20 q^{82} -3.05240e17 q^{83} -1.78885e19 q^{84} -1.58669e20 q^{86} +1.63370e20 q^{87} -1.95835e20 q^{88} +2.34593e20 q^{89} +4.66787e20 q^{91} +2.15162e19 q^{92} -4.93917e20 q^{93} -5.45116e20 q^{94} -1.68239e20 q^{96} +5.92086e20 q^{97} -2.02274e20 q^{98} +2.08365e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 666 q^{2} - 118098 q^{3} + 1179236 q^{4} + 39326634 q^{6} - 679896112 q^{7} - 2427055848 q^{8} + 6973568802 q^{9}+O(q^{10}) $ 2 * q - 666 * q^2 - 118098 * q^3 + 1179236 * q^4 + 39326634 * q^6 - 679896112 * q^7 - 2427055848 * q^8 + 6973568802 * q^9	$ 2 q - 666 q^{2} - 118098 q^{3} + 1179236 q^{4} + 39326634 q^{6} - 679896112 q^{7} - 2427055848 q^{8} + 6973568802 q^{9} + 219869122968 q^{11} - 69632706564 q^{12} + 48468909956 q^{13} - 711297706896 q^{14} - 8288736440560 q^{16} + 11333529041436 q^{17} - 2322198411066 q^{18} + 11960585011624 q^{19} + 40147185517488 q^{21} - 234277148563128 q^{22} + 146508390063504 q^{23} + 143315220768552 q^{24} - 24\!\cdots\!16 q^{26}+ \cdots + 76\!\cdots\!68 q^{99}+O(q^{100}) $ 2 * q - 666 * q^2 - 118098 * q^3 + 1179236 * q^4 + 39326634 * q^6 - 679896112 * q^7 - 2427055848 * q^8 + 6973568802 * q^9 + 219869122968 * q^11 - 69632706564 * q^12 + 48468909956 * q^13 - 711297706896 * q^14 - 8288736440560 * q^16 + 11333529041436 * q^17 - 2322198411066 * q^18 + 11960585011624 * q^19 + 40147185517488 * q^21 - 234277148563128 * q^22 + 146508390063504 * q^23 + 143315220768552 * q^24 - 2464447363969716 * q^26 - 411782264189298 * q^27 + 223631286954656 * q^28 - 1798520043674052 * q^29 + 11169107526944992 * q^31 + 10999491515934048 * q^32 - 12983051842137432 * q^33 + 4867007487413652 * q^34 + 4111741689897636 * q^36 - 12736264858660012 * q^37 + 111335877667820664 * q^38 - 2862040663991844 * q^39 + 122972020616468052 * q^41 + 42001418294501904 * q^42 - 288455418162270040 * q^43 + 236905239135592368 * q^44 - 428027981395161168 * q^46 - 837243745741596960 * q^47 + 489441598078627440 * q^48 - 715285396941470670 * q^49 - 669233556367754364 * q^51 + 1659150748242045896 * q^52 + 43007964012775764 * q^53 + 137123493975036234 * q^54 + 2030730320050568640 * q^56 - 706260584351385576 * q^57 - 5395375030104291852 * q^58 - 3523823330903857224 * q^59 - 1779023128451013860 * q^61 + 5204128043677477536 * q^62 - 2370651157622148912 * q^63 + 5211692013463482944 * q^64 + 13833831345504145272 * q^66 + 16454068667621610296 * q^67 + 927498335984326008 * q^68 - 8651173924859847696 * q^69 + 17379227131150420944 * q^71 - 8462620471162227048 * q^72 - 50891146268473989076 * q^73 + 81714426393675287484 * q^74 - 69750112936092790064 * q^76 - 45428450983816025664 * q^77 + 145523152395047760084 * q^78 - 54055785594190591040 * q^79 + 24315330918113857602 * q^81 + 227933719179803895996 * q^82 - 111108429277666677288 * q^83 - 13205203863385482144 * q^84 + 158596867265246585208 * q^86 + 106200810058909096548 * q^87 - 59732519594715541728 * q^88 + 226920767965448065524 * q^89 + 429154174685185000352 * q^91 + 338958281810136649248 * q^92 - 659524630358574832608 * q^93 + 246883115968835119776 * q^94 - 649508974524389600352 * q^96 + 128331469252795746236 * q^97 + 875730737371150330614 * q^98 + 766636228226373222168 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1271.96	0.878328	0.439164	−	0.898407i	$-0.355275\pi$
$2$	1271.96	0.878328	0.439164	+	0.898407i	$0.355275\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	−479282.	−0.228540
$5$	0	0
$5$	0	0
$6$	−7.51077e7	−0.507103
$7$	−6.32076e8	−0.845745	−0.422873	−	0.906189i	$-0.638978\pi$
$7$	−6.32076e8	−0.845745	−0.422873	+	0.906189i	$0.638978\pi$
$8$	−3.27711e9	−1.07906
$9$	3.48678e9	0.333333
$10$	0	0
$11$	5.97585e10	0.694666	0.347333	−	0.937742i	$-0.387087\pi$
$11$	5.97585e10	0.694666	0.347333	+	0.937742i	$0.387087\pi$
$12$	2.83011e10	0.131947
$13$	−7.38499e11	−1.48575	−0.742874	−	0.669432i	$-0.766538\pi$
$13$	−7.38499e11	−1.48575	−0.742874	+	0.669432i	$0.766538\pi$
$14$	−8.03972e11	−0.742842
$15$	0	0
$16$	−3.16321e12	−0.719230
$17$	8.35876e12	1.00561	0.502804	−	0.864401i	$-0.332302\pi$
$17$	8.35876e12	1.00561	0.502804	+	0.864401i	$0.332302\pi$
$18$	4.43503e12	0.292776
$19$	4.19061e13	1.56807	0.784034	−	0.620718i	$-0.213159\pi$
$19$	4.19061e13	1.56807	0.784034	+	0.620718i	$0.213159\pi$
$20$	0	0
$21$	3.73234e13	0.488291
$22$	7.60101e13	0.610145
$23$	−4.48926e13	−0.225961	−0.112980	−	0.993597i	$-0.536040\pi$
$23$	−4.48926e13	−0.225961	−0.112980	+	0.993597i	$0.536040\pi$
$24$	1.93510e14	0.622996
$25$	0	0
$26$	−9.39338e14	−1.30497
$27$	−2.05891e14	−0.192450
$28$	3.02943e14	0.193286
$29$	−2.76669e15	−1.22119	−0.610593	−	0.791945i	$-0.709069\pi$
$29$	−2.76669e15	−1.22119	−0.610593	+	0.791945i	$0.709069\pi$
$30$	0	0
$31$	8.36452e15	1.83292	0.916459	−	0.400128i	$-0.131034\pi$
$31$	8.36452e15	1.83292	0.916459	+	0.400128i	$0.131034\pi$
$32$	2.84914e15	0.447341
$33$	−3.52868e15	−0.401066
$34$	1.06320e16	0.883253
$35$	0	0
$36$	−1.67115e15	−0.0761798
$37$	1.77675e16	0.607447	0.303724	−	0.952760i	$-0.401770\pi$
$37$	1.77675e16	0.607447	0.303724	+	0.952760i	$0.401770\pi$
$38$	5.33027e16	1.37728
$39$	4.36077e16	0.857797
$40$	0	0
$41$	1.45253e17	1.69003	0.845013	−	0.534745i	$-0.179592\pi$
$41$	1.45253e17	1.69003	0.845013	+	0.534745i	$0.179592\pi$
$42$	4.74737e16	0.428880
$43$	−1.24744e17	−0.880239	−0.440119	−	0.897939i	$-0.645064\pi$
$43$	−1.24744e17	−0.880239	−0.440119	+	0.897939i	$0.645064\pi$
$44$	−2.86412e16	−0.158759
$45$	0	0
$46$	−5.71014e16	−0.198468
$47$	−4.28566e17	−1.18847	−0.594237	−	0.804290i	$-0.702546\pi$
$47$	−4.28566e17	−1.18847	−0.594237	+	0.804290i	$0.702546\pi$
$48$	1.86784e17	0.415248
$49$	−1.59026e17	−0.284715
$50$	0	0
$51$	−4.93577e17	−0.580588
$52$	3.53950e17	0.339552
$53$	4.77017e17	0.374660	0.187330	−	0.982297i	$-0.440017\pi$
$53$	4.77017e17	0.374660	0.187330	+	0.982297i	$0.440017\pi$
$54$	−2.61884e17	−0.169034
$55$	0	0
$56$	2.07138e18	0.912611
$57$	−2.47452e18	−0.905324
$58$	−3.51911e18	−1.07260
$59$	1.61959e18	0.412533	0.206267	−	0.978496i	$-0.433869\pi$
$59$	1.61959e18	0.412533	0.206267	+	0.978496i	$0.433869\pi$
$60$	0	0
$61$	−3.76882e18	−0.676461	−0.338230	−	0.941063i	$-0.609828\pi$
$61$	−3.76882e18	−0.676461	−0.338230	+	0.941063i	$0.609828\pi$
$62$	1.06393e19	1.60990
$63$	−2.20391e18	−0.281915
$64$	1.02577e19	1.11214
$65$	0	0
$66$	−4.48832e18	−0.352267
$67$	2.81797e18	0.188865	0.0944324	−	0.995531i	$-0.469896\pi$
$67$	2.81797e18	0.188865	0.0944324	+	0.995531i	$0.469896\pi$
$68$	−4.00621e18	−0.229821
$69$	2.65086e18	0.130458
$70$	0	0
$71$	1.00228e19	0.365407	0.182704	−	0.983168i	$-0.441515\pi$
$71$	1.00228e19	0.365407	0.182704	+	0.983168i	$0.441515\pi$
$72$	−1.14266e19	−0.359687
$73$	1.72739e19	0.470435	0.235218	−	0.971943i	$-0.424420\pi$
$73$	1.72739e19	0.470435	0.235218	+	0.971943i	$0.424420\pi$
$74$	2.25995e19	0.533538
$75$	0	0
$76$	−2.00849e19	−0.358365
$77$	−3.77719e19	−0.587511
$78$	5.54670e19	0.753427
$79$	−3.28276e19	−0.390080	−0.195040	−	0.980795i	$-0.562484\pi$
$79$	−3.28276e19	−0.390080	−0.195040	+	0.980795i	$0.562484\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	1.84755e20	1.48440
$83$	−3.05240e17	−0.00215934	−0.00107967	−	0.999999i	$-0.500344\pi$
$83$	−3.05240e17	−0.00215934	−0.00107967	+	0.999999i	$0.500344\pi$
$84$	−1.78885e19	−0.111594
$85$	0	0
$86$	−1.58669e20	−0.773139
$87$	1.63370e20	0.705052
$88$	−1.95835e20	−0.749587
$89$	2.34593e20	0.797480	0.398740	−	0.917064i	$-0.369448\pi$
$89$	2.34593e20	0.797480	0.398740	+	0.917064i	$0.369448\pi$
$90$	0	0
$91$	4.66787e20	1.25656
$92$	2.15162e19	0.0516409
$93$	−4.93917e20	−1.05824
$94$	−5.45116e20	−1.04387
$95$	0	0
$96$	−1.68239e20	−0.258272
$97$	5.92086e20	0.815233	0.407616	−	0.913153i	$-0.366360\pi$
$97$	5.92086e20	0.815233	0.407616	+	0.913153i	$0.366360\pi$
$98$	−2.02274e20	−0.250073
$99$	2.08365e20	0.231555
$100$	0	0
$101$	−1.66229e21	−1.49739	−0.748693	−	0.662917i	$-0.769318\pi$
$101$	−1.66229e21	−1.49739	−0.748693	+	0.662917i	$0.769318\pi$
$102$	−6.27807e20	−0.509946
$103$	−1.17919e21	−0.864552	−0.432276	−	0.901741i	$-0.642289\pi$
$103$	−1.17919e21	−0.864552	−0.432276	+	0.901741i	$0.642289\pi$
$104$	2.42014e21	1.60321
$105$	0	0
$106$	6.06745e20	0.329075
$107$	8.69160e20	0.427140	0.213570	−	0.976928i	$-0.431491\pi$
$107$	8.69160e20	0.427140	0.213570	+	0.976928i	$0.431491\pi$
$108$	9.86799e19	0.0439825
$109$	4.12021e20	0.166702	0.0833510	−	0.996520i	$-0.473438\pi$
$109$	4.12021e20	0.166702	0.0833510	+	0.996520i	$0.473438\pi$
$110$	0	0
$111$	−1.04915e21	−0.350710
$112$	1.99939e21	0.608286
$113$	−2.45943e21	−0.681569	−0.340785	−	0.940141i	$-0.610693\pi$
$113$	−2.45943e21	−0.681569	−0.340785	+	0.940141i	$0.610693\pi$
$114$	−3.14747e21	−0.795172
$115$	0	0
$116$	1.32602e21	0.279089
$117$	−2.57499e21	−0.495249
$118$	2.06005e21	0.362340
$119$	−5.28337e21	−0.850487
$120$	0	0
$121$	−3.82917e21	−0.517439
$122$	−4.79378e21	−0.594155
$123$	−8.57702e21	−0.975737
$124$	−4.00896e21	−0.418894
$125$	0	0
$126$	−2.80328e21	−0.247614
$127$	−5.53783e21	−0.450195	−0.225097	−	0.974336i	$-0.572270\pi$
$127$	−5.53783e21	−0.450195	−0.225097	+	0.974336i	$0.572270\pi$
$128$	7.07226e21	0.529485
$129$	7.36600e21	0.508206
$130$	0	0
$131$	−2.54630e22	−1.49472	−0.747362	−	0.664417i	$-0.768680\pi$
$131$	−2.54630e22	−1.49472	−0.747362	+	0.664417i	$0.768680\pi$
$132$	1.69123e21	0.0916594
$133$	−2.64878e22	−1.32619
$134$	3.58433e21	0.165885
$135$	0	0
$136$	−2.73926e22	−1.08511
$137$	4.82088e22	1.76832	0.884158	−	0.467188i	$-0.154733\pi$
$137$	4.82088e22	1.76832	0.884158	+	0.467188i	$0.154733\pi$
$138$	3.37178e21	0.114585
$139$	−2.18194e22	−0.687363	−0.343682	−	0.939086i	$-0.611674\pi$
$139$	−2.18194e22	−0.687363	−0.343682	+	0.939086i	$0.611674\pi$
$140$	0	0
$141$	2.53064e22	0.686166
$142$	1.27486e22	0.320948
$143$	−4.41316e22	−1.03210
$144$	−1.10294e22	−0.239743
$145$	0	0
$146$	2.19716e22	0.413197
$147$	9.39035e21	0.164380
$148$	−8.51565e21	−0.138826
$149$	4.84140e22	0.735386	0.367693	−	0.929947i	$-0.380148\pi$
$149$	4.84140e22	0.735386	0.367693	+	0.929947i	$0.380148\pi$
$150$	0	0
$151$	−1.22948e23	−1.62354	−0.811770	−	0.583977i	$-0.801496\pi$
$151$	−1.22948e23	−1.62354	−0.811770	+	0.583977i	$0.801496\pi$
$152$	−1.37331e23	−1.69204
$153$	2.91452e22	0.335202
$154$	−4.80441e22	−0.516027
$155$	0	0
$156$	−2.09004e22	−0.196040
$157$	−5.89828e22	−0.517344	−0.258672	−	0.965965i	$-0.583285\pi$
$157$	−5.89828e22	−0.517344	−0.258672	+	0.965965i	$0.583285\pi$
$158$	−4.17552e22	−0.342619
$159$	−2.81674e22	−0.216310
$160$	0	0
$161$	2.83755e22	0.191105
$162$	1.54640e22	0.0975920
$163$	3.18244e22	0.188274	0.0941369	−	0.995559i	$-0.469991\pi$
$163$	3.18244e22	0.188274	0.0941369	+	0.995559i	$0.469991\pi$
$164$	−6.96170e22	−0.386238
$165$	0	0
$166$	−3.88252e20	−0.00189661
$167$	3.62822e22	0.166407	0.0832033	−	0.996533i	$-0.473485\pi$
$167$	3.62822e22	0.166407	0.0832033	+	0.996533i	$0.473485\pi$
$168$	−1.22313e23	−0.526896
$169$	2.98317e23	1.20744
$170$	0	0
$171$	1.46118e23	0.522689
$172$	5.97875e22	0.201169
$173$	−3.21510e23	−1.01791	−0.508956	−	0.860792i	$-0.669969\pi$
$173$	−3.21510e23	−1.01791	−0.508956	+	0.860792i	$0.669969\pi$
$174$	2.07800e23	0.619267
$175$	0	0
$176$	−1.89028e23	−0.499625
$177$	−9.56351e22	−0.238176
$178$	2.98392e23	0.700449
$179$	−1.80418e23	−0.399321	−0.199661	−	0.979865i	$-0.563984\pi$
$179$	−1.80418e23	−0.399321	−0.199661	+	0.979865i	$0.563984\pi$
$180$	0	0
$181$	−6.36646e21	−0.0125393	−0.00626965	−	0.999980i	$-0.501996\pi$
$181$	−6.36646e21	−0.0125393	−0.00626965	+	0.999980i	$0.501996\pi$
$182$	5.93733e23	1.10368
$183$	2.22545e23	0.390555
$184$	1.47118e23	0.243825
$185$	0	0
$186$	−6.28240e23	−0.929479
$187$	4.99507e23	0.698561
$188$	2.05404e23	0.271613
$189$	1.30139e23	0.162764
$190$	0	0
$191$	8.17882e23	0.915883	0.457942	−	0.888982i	$-0.348587\pi$
$191$	8.17882e23	0.915883	0.457942	+	0.888982i	$0.348587\pi$
$192$	−6.05707e23	−0.642096
$193$	1.80064e23	0.180748	0.0903742	−	0.995908i	$-0.471194\pi$
$193$	1.80064e23	0.180748	0.0903742	+	0.995908i	$0.471194\pi$
$194$	7.53107e23	0.716042
$195$	0	0
$196$	7.62185e22	0.0650686
$197$	−1.65216e24	−1.33708	−0.668541	−	0.743675i	$-0.733081\pi$
$197$	−1.65216e24	−1.33708	−0.668541	+	0.743675i	$0.733081\pi$
$198$	2.65031e23	0.203382
$199$	−1.52537e24	−1.11024	−0.555120	−	0.831770i	$-0.687328\pi$
$199$	−1.52537e24	−1.11024	−0.555120	+	0.831770i	$0.687328\pi$
$200$	0	0
$201$	−1.66398e23	−0.109041
$202$	−2.11436e24	−1.31520
$203$	1.74876e24	1.03281
$204$	2.36562e23	0.132687
$205$	0	0
$206$	−1.49987e24	−0.759361
$207$	−1.56531e23	−0.0753202
$208$	2.33603e24	1.06859
$209$	2.50425e24	1.08928
$210$	0	0
$211$	1.35748e24	0.534277	0.267138	−	0.963658i	$-0.413922\pi$
$211$	1.35748e24	0.534277	0.267138	+	0.963658i	$0.413922\pi$
$212$	−2.28626e23	−0.0856247
$213$	−5.91838e23	−0.210968
$214$	1.10553e24	0.375169
$215$	0	0
$216$	6.74728e23	0.207665
$217$	−5.28701e24	−1.55018
$218$	5.24072e23	0.146419
$219$	−1.02001e24	−0.271606
$220$	0	0
$221$	−6.17294e24	−1.49408
$222$	−1.33448e24	−0.308038
$223$	−4.73192e24	−1.04192	−0.520962	−	0.853580i	$-0.674427\pi$
$223$	−4.73192e24	−1.04192	−0.520962	+	0.853580i	$0.674427\pi$
$224$	−1.80087e24	−0.378336
$225$	0	0
$226$	−3.12828e24	−0.598642
$227$	−8.00226e24	−1.46198	−0.730989	−	0.682389i	$-0.760941\pi$
$227$	−8.00226e24	−1.46198	−0.730989	+	0.682389i	$0.760941\pi$
$228$	1.18599e24	0.206902
$229$	2.72111e24	0.453391	0.226696	−	0.973966i	$-0.427208\pi$
$229$	2.72111e24	0.453391	0.226696	+	0.973966i	$0.427208\pi$
$230$	0	0
$231$	2.23039e24	0.339200
$232$	9.06674e24	1.31773
$233$	6.60115e24	0.917028	0.458514	−	0.888687i	$-0.348382\pi$
$233$	6.60115e24	0.917028	0.458514	+	0.888687i	$0.348382\pi$
$234$	−3.27527e24	−0.434991
$235$	0	0
$236$	−7.76240e23	−0.0942801
$237$	1.93843e24	0.225213
$238$	−6.72021e24	−0.747007
$239$	9.92224e24	1.05544	0.527718	−	0.849420i	$-0.323048\pi$
$239$	9.92224e24	1.05544	0.527718	+	0.849420i	$0.323048\pi$
$240$	0	0
$241$	−1.54177e25	−1.50259	−0.751295	−	0.659966i	$-0.770571\pi$
$241$	−1.54177e25	−1.50259	−0.751295	+	0.659966i	$0.770571\pi$
$242$	−4.87054e24	−0.454481
$243$	−7.17898e23	−0.0641500
$244$	1.80633e24	0.154598
$245$	0	0
$246$	−1.09096e25	−0.857018
$247$	−3.09477e25	−2.32975
$248$	−2.74114e25	−1.97783
$249$	1.80241e22	0.00124670
$250$	0	0
$251$	−1.00009e25	−0.636014	−0.318007	−	0.948088i	$-0.603014\pi$
$251$	−1.00009e25	−0.636014	−0.318007	+	0.948088i	$0.603014\pi$
$252$	1.05630e24	0.0644287
$253$	−2.68271e24	−0.156967
$254$	−7.04387e24	−0.395419
$255$	0	0
$256$	−1.25164e25	−0.647080
$257$	3.06596e25	1.52149	0.760744	−	0.649052i	$-0.224834\pi$
$257$	3.06596e25	1.52149	0.760744	+	0.649052i	$0.224834\pi$
$258$	9.36923e24	0.446372
$259$	−1.12304e25	−0.513746
$260$	0	0
$261$	−9.64685e24	−0.407062
$262$	−3.23878e25	−1.31286
$263$	2.33455e25	0.909216	0.454608	−	0.890692i	$-0.349779\pi$
$263$	2.33455e25	0.909216	0.454608	+	0.890692i	$0.349779\pi$
$264$	1.15639e25	0.432775
$265$	0	0
$266$	−3.36914e25	−1.16483
$267$	−1.38525e25	−0.460425
$268$	−1.35060e24	−0.0431631
$269$	2.29737e25	0.706043	0.353022	−	0.935615i	$-0.385154\pi$
$269$	2.29737e25	0.706043	0.353022	+	0.935615i	$0.385154\pi$
$270$	0	0
$271$	−4.02085e25	−1.14325	−0.571624	−	0.820516i	$-0.693686\pi$
$271$	−4.02085e25	−1.14325	−0.571624	+	0.820516i	$0.693686\pi$
$272$	−2.64405e25	−0.723263
$273$	−2.75633e25	−0.725477
$274$	6.13194e25	1.55316
$275$	0	0
$276$	−1.27051e24	−0.0298149
$277$	4.48722e25	1.01377	0.506886	−	0.862013i	$-0.330797\pi$
$277$	4.48722e25	1.01377	0.506886	+	0.862013i	$0.330797\pi$
$278$	−2.77532e25	−0.603731
$279$	2.91653e25	0.610973
$280$	0	0
$281$	−6.06489e25	−1.17871	−0.589355	−	0.807874i	$-0.700618\pi$
$281$	−6.06489e25	−1.17871	−0.589355	+	0.807874i	$0.700618\pi$
$282$	3.21886e25	0.602679
$283$	−1.04229e26	−1.88032	−0.940158	−	0.340739i	$-0.889323\pi$
$283$	−1.04229e26	−1.88032	−0.940158	+	0.340739i	$0.889323\pi$
$284$	−4.80376e24	−0.0835100
$285$	0	0
$286$	−5.61334e25	−0.906521
$287$	−9.18107e25	−1.42933
$288$	9.93433e24	0.149114
$289$	7.76980e23	0.0112456
$290$	0	0
$291$	−3.49621e25	−0.470675
$292$	−8.27907e24	−0.107513
$293$	−6.24634e25	−0.782556	−0.391278	−	0.920273i	$-0.627967\pi$
$293$	−6.24634e25	−0.782556	−0.391278	+	0.920273i	$0.627967\pi$
$294$	1.19441e25	0.144380
$295$	0	0
$296$	−5.82261e25	−0.655472
$297$	−1.23037e25	−0.133689
$298$	6.15805e25	0.645911
$299$	3.31532e25	0.335720
$300$	0	0
$301$	7.88476e25	0.744458
$302$	−1.56384e26	−1.42600
$303$	9.81568e25	0.864516
$304$	−1.32558e26	−1.12780
$305$	0	0
$306$	3.70714e25	0.294418
$307$	−7.14411e25	−0.548271	−0.274135	−	0.961691i	$-0.588392\pi$
$307$	−7.14411e25	−0.548271	−0.274135	+	0.961691i	$0.588392\pi$
$308$	1.81034e25	0.134269
$309$	6.96297e25	0.499150
$310$	0	0
$311$	−6.01915e25	−0.403228	−0.201614	−	0.979465i	$-0.564619\pi$
$311$	−6.01915e25	−0.403228	−0.201614	+	0.979465i	$0.564619\pi$
$312$	−1.42907e26	−0.925615
$313$	1.69049e26	1.05876	0.529379	−	0.848385i	$-0.322425\pi$
$313$	1.69049e26	1.05876	0.529379	+	0.848385i	$0.322425\pi$
$314$	−7.50235e25	−0.454398
$315$	0	0
$316$	1.57337e25	0.0891488
$317$	5.71691e25	0.313357	0.156678	−	0.987650i	$-0.449921\pi$
$317$	5.71691e25	0.313357	0.156678	+	0.987650i	$0.449921\pi$
$318$	−3.58277e25	−0.189991
$319$	−1.65333e26	−0.848316
$320$	0	0
$321$	−5.13230e25	−0.246609
$322$	3.60924e25	0.167853
$323$	3.50283e26	1.57686
$324$	−5.82695e24	−0.0253933
$325$	0	0
$326$	4.04792e25	0.165366
$327$	−2.43294e25	−0.0962454
$328$	−4.76009e26	−1.82364
$329$	2.70886e26	1.00515
$330$	0	0
$331$	6.67957e25	0.232571	0.116285	−	0.993216i	$-0.462901\pi$
$331$	6.67957e25	0.232571	0.116285	+	0.993216i	$0.462901\pi$
$332$	1.46296e23	0.000493495 0
$333$	6.19515e25	0.202482
$334$	4.61493e25	0.146160
$335$	0	0
$336$	−1.18062e26	−0.351194
$337$	−3.55365e26	−1.02461	−0.512307	−	0.858802i	$-0.671209\pi$
$337$	−3.55365e26	−1.02461	−0.512307	+	0.858802i	$0.671209\pi$
$338$	3.79446e26	1.06053
$339$	1.45227e26	0.393504
$340$	0	0
$341$	4.99851e26	1.27327
$342$	1.85855e26	0.459093
$343$	4.53560e26	1.08654
$344$	4.08799e26	0.949832
$345$	0	0
$346$	−4.08947e26	−0.894062
$347$	3.94238e25	0.0836179	0.0418090	−	0.999126i	$-0.486688\pi$
$347$	3.94238e25	0.0836179	0.0418090	+	0.999126i	$0.486688\pi$
$348$	−7.83004e25	−0.161132
$349$	−5.84299e26	−1.16672	−0.583362	−	0.812212i	$-0.698263\pi$
$349$	−5.84299e26	−1.16672	−0.583362	+	0.812212i	$0.698263\pi$
$350$	0	0
$351$	1.52050e26	0.285932
$352$	1.70260e26	0.310753
$353$	−4.96062e26	−0.878823	−0.439412	−	0.898286i	$-0.644813\pi$
$353$	−4.96062e26	−0.878823	−0.439412	+	0.898286i	$0.644813\pi$
$354$	−1.21644e26	−0.209197
$355$	0	0
$356$	−1.12436e26	−0.182256
$357$	3.11978e26	0.491029
$358$	−2.29483e26	−0.350735
$359$	−3.35136e26	−0.497426	−0.248713	−	0.968577i	$-0.580008\pi$
$359$	−3.35136e26	−0.497426	−0.248713	+	0.968577i	$0.580008\pi$
$360$	0	0
$361$	1.04192e27	1.45884
$362$	−8.09785e24	−0.0110136
$363$	2.26109e26	0.298743
$364$	−2.23723e26	−0.287174
$365$	0	0
$366$	2.83068e26	0.343035
$367$	−9.70198e25	−0.114253	−0.0571264	−	0.998367i	$-0.518194\pi$
$367$	−9.70198e25	−0.114253	−0.0571264	+	0.998367i	$0.518194\pi$
$368$	1.42005e26	0.162518
$369$	5.06465e26	0.563342
$370$	0	0
$371$	−3.01511e26	−0.316867
$372$	2.36725e26	0.241849
$373$	−1.03252e27	−1.02554	−0.512771	−	0.858525i	$-0.671381\pi$
$373$	−1.03252e27	−1.02554	−0.512771	+	0.858525i	$0.671381\pi$
$374$	6.35350e26	0.613566
$375$	0	0
$376$	1.40446e27	1.28244
$377$	2.04320e27	1.81437
$378$	1.65531e26	0.142960
$379$	2.20881e26	0.185544	0.0927721	−	0.995687i	$-0.470427\pi$
$379$	2.20881e26	0.185544	0.0927721	+	0.995687i	$0.470427\pi$
$380$	0	0
$381$	3.27003e26	0.259920
$382$	1.04031e27	0.804446
$383$	1.59547e27	1.20033	0.600165	−	0.799876i	$-0.295101\pi$
$383$	1.59547e27	1.20033	0.600165	+	0.799876i	$0.295101\pi$
$384$	−4.17610e26	−0.305698
$385$	0	0
$386$	2.29033e26	0.158756
$387$	−4.34955e26	−0.293413
$388$	−2.83776e26	−0.186313
$389$	−3.60474e26	−0.230358	−0.115179	−	0.993345i	$-0.536744\pi$
$389$	−3.60474e26	−0.230358	−0.115179	+	0.993345i	$0.536744\pi$
$390$	0	0
$391$	−3.75247e26	−0.227228
$392$	5.21147e26	0.307225
$393$	1.50357e27	0.862980
$394$	−2.10148e27	−1.17440
$395$	0	0
$396$	−9.98656e25	−0.0529196
$397$	1.47191e27	0.759595	0.379798	−	0.925070i	$-0.375994\pi$
$397$	1.47191e27	0.759595	0.379798	+	0.925070i	$0.375994\pi$
$398$	−1.94020e27	−0.975156
$399$	1.56408e27	0.765674
$400$	0	0
$401$	1.00401e27	0.466360	0.233180	−	0.972434i	$-0.425087\pi$
$401$	1.00401e27	0.466360	0.233180	+	0.972434i	$0.425087\pi$
$402$	−2.11651e26	−0.0957740
$403$	−6.17719e27	−2.72325
$404$	7.96708e26	0.342212
$405$	0	0
$406$	2.22434e27	0.907148
$407$	1.06176e27	0.421973
$408$	1.61750e27	0.626489
$409$	−4.41838e27	−1.66789	−0.833947	−	0.551845i	$-0.813924\pi$
$409$	−4.41838e27	−1.66789	−0.833947	+	0.551845i	$0.813924\pi$
$410$	0	0
$411$	−2.84668e27	−1.02094
$412$	5.65163e26	0.197584
$413$	−1.02370e27	−0.348898
$414$	−1.99100e26	−0.0661559
$415$	0	0
$416$	−2.10409e27	−0.664635
$417$	1.28841e27	0.396849
$418$	3.18529e27	0.956749
$419$	−2.36639e27	−0.693169	−0.346584	−	0.938019i	$-0.612659\pi$
$419$	−2.36639e27	−0.693169	−0.346584	+	0.938019i	$0.612659\pi$
$420$	0	0
$421$	2.72067e27	0.758078	0.379039	−	0.925381i	$-0.376255\pi$
$421$	2.72067e27	0.758078	0.379039	+	0.925381i	$0.376255\pi$
$422$	1.72665e27	0.469271
$423$	−1.49432e27	−0.396158
$424$	−1.56324e27	−0.404281
$425$	0	0
$426$	−7.52791e26	−0.185299
$427$	2.38218e27	0.572114
$428$	−4.16573e26	−0.0976183
$429$	2.60593e27	0.595883
$430$	0	0
$431$	8.87397e27	1.93244	0.966222	−	0.257713i	$-0.0829688\pi$
$431$	8.87397e27	1.93244	0.966222	+	0.257713i	$0.0829688\pi$
$432$	6.51276e26	0.138416
$433$	3.39994e27	0.705258	0.352629	−	0.935763i	$-0.385288\pi$
$433$	3.39994e27	0.705258	0.352629	+	0.935763i	$0.385288\pi$
$434$	−6.72484e27	−1.36157
$435$	0	0
$436$	−1.97474e26	−0.0380980
$437$	−1.88128e27	−0.354322
$438$	−1.29740e27	−0.238559
$439$	2.29516e27	0.412037	0.206018	−	0.978548i	$-0.433949\pi$
$439$	2.29516e27	0.412037	0.206018	+	0.978548i	$0.433949\pi$
$440$	0	0
$441$	−5.54491e26	−0.0949050
$442$	−7.85170e27	−1.31229
$443$	5.65265e27	0.922599	0.461299	−	0.887245i	$-0.347383\pi$
$443$	5.65265e27	0.922599	0.461299	+	0.887245i	$0.347383\pi$
$444$	5.02841e26	0.0801510
$445$	0	0
$446$	−6.01879e27	−0.915152
$447$	−2.85880e27	−0.424576
$448$	−6.48364e27	−0.940589
$449$	1.23324e28	1.74768	0.873839	−	0.486216i	$-0.161623\pi$
$449$	1.23324e28	1.74768	0.873839	+	0.486216i	$0.161623\pi$
$450$	0	0
$451$	8.68008e27	1.17400
$452$	1.17876e27	0.155766
$453$	7.25995e27	0.937351
$454$	−1.01785e28	−1.28410
$455$	0	0
$456$	8.10926e27	0.976900
$457$	7.62646e27	0.897848	0.448924	−	0.893570i	$-0.351807\pi$
$457$	7.62646e27	0.897848	0.448924	+	0.893570i	$0.351807\pi$
$458$	3.46113e27	0.398226
$459$	−1.72100e27	−0.193529
$460$	0	0
$461$	−9.04851e26	−0.0972114	−0.0486057	−	0.998818i	$-0.515478\pi$
$461$	−9.04851e26	−0.0972114	−0.0486057	+	0.998818i	$0.515478\pi$
$462$	2.83696e27	0.297929
$463$	1.19705e27	0.122889	0.0614443	−	0.998111i	$-0.480429\pi$
$463$	1.19705e27	0.122889	0.0614443	+	0.998111i	$0.480429\pi$
$464$	8.75161e27	0.878313
$465$	0	0
$466$	8.39637e27	0.805452
$467$	1.16951e28	1.09693	0.548464	−	0.836174i	$-0.315213\pi$
$467$	1.16951e28	1.09693	0.548464	+	0.836174i	$0.315213\pi$
$468$	1.23415e27	0.113184
$469$	−1.78117e27	−0.159732
$470$	0	0
$471$	3.48288e27	0.298689
$472$	−5.30757e27	−0.445148
$473$	−7.45451e27	−0.611472
$474$	2.46560e27	0.197811
$475$	0	0
$476$	2.53222e27	0.194370
$477$	1.66326e27	0.124887
$478$	1.26206e28	0.927019
$479$	2.04856e28	1.47206	0.736029	−	0.676950i	$-0.236699\pi$
$479$	2.04856e28	1.47206	0.736029	+	0.676950i	$0.236699\pi$
$480$	0	0
$481$	−1.31213e28	−0.902513
$482$	−1.96106e28	−1.31977
$483$	−1.67555e27	−0.110335
$484$	1.83525e27	0.118255
$485$	0	0
$486$	−9.13134e26	−0.0563448
$487$	3.75693e27	0.226871	0.113436	−	0.993545i	$-0.463814\pi$
$487$	3.75693e27	0.226871	0.113436	+	0.993545i	$0.463814\pi$
$488$	1.23508e28	0.729942
$489$	−1.87920e27	−0.108700
$490$	0	0
$491$	−1.53173e28	−0.848841	−0.424420	−	0.905465i	$-0.639522\pi$
$491$	−1.53173e28	−0.848841	−0.424420	+	0.905465i	$0.639522\pi$
$492$	4.11081e27	0.222995
$493$	−2.31261e28	−1.22803
$494$	−3.93640e28	−2.04629
$495$	0	0
$496$	−2.64587e28	−1.31829
$497$	−6.33518e27	−0.309042
$498$	2.29259e25	0.00109501
$499$	3.84836e27	0.179978	0.0899890	−	0.995943i	$-0.471317\pi$
$499$	3.84836e27	0.179978	0.0899890	+	0.995943i	$0.471317\pi$
$500$	0	0
$501$	−2.14243e27	−0.0960749
$502$	−1.27208e28	−0.558629
$503$	2.25177e28	0.968411	0.484205	−	0.874954i	$-0.339109\pi$
$503$	2.25177e28	0.968411	0.484205	+	0.874954i	$0.339109\pi$
$504$	7.22246e27	0.304204
$505$	0	0
$506$	−3.41229e27	−0.137869
$507$	−1.76153e28	−0.697119
$508$	2.65418e27	0.102887
$509$	2.09122e27	0.0794078	0.0397039	−	0.999211i	$-0.487359\pi$
$509$	2.09122e27	0.0794078	0.0397039	+	0.999211i	$0.487359\pi$
$510$	0	0
$511$	−1.09184e28	−0.397869
$512$	−3.07519e28	−1.09783
$513$	−8.62810e27	−0.301775
$514$	3.89976e28	1.33637
$515$	0	0
$516$	−3.53039e27	−0.116145
$517$	−2.56104e28	−0.825594
$518$	−1.42846e28	−0.451237
$519$	1.89849e28	0.587692
$520$	0	0
$521$	−3.76486e27	−0.111932	−0.0559659	−	0.998433i	$-0.517824\pi$
$521$	−3.76486e27	−0.111932	−0.0559659	+	0.998433i	$0.517824\pi$
$522$	−1.22704e28	−0.357534
$523$	1.74944e28	0.499611	0.249806	−	0.968296i	$-0.419633\pi$
$523$	1.74944e28	0.499611	0.249806	+	0.968296i	$0.419633\pi$
$524$	1.22040e28	0.341604
$525$	0	0
$526$	2.96944e28	0.798590
$527$	6.99170e28	1.84320
$528$	1.11619e28	0.288459
$529$	−3.74562e28	−0.948942
$530$	0	0
$531$	5.64716e27	0.137511
$532$	1.26952e28	0.303086
$533$	−1.07269e29	−2.51095
$534$	−1.76197e28	−0.404404
$535$	0	0
$536$	−9.23480e27	−0.203797
$537$	1.06535e28	0.230548
$538$	2.92215e28	0.620138
$539$	−9.50317e27	−0.197782
$540$	0	0
$541$	−6.99960e28	−1.40121	−0.700603	−	0.713552i	$-0.747085\pi$
$541$	−6.99960e28	−1.40121	−0.700603	+	0.713552i	$0.747085\pi$
$542$	−5.11434e28	−1.00415
$543$	3.75933e26	0.00723956
$544$	2.38153e28	0.449849
$545$	0	0
$546$	−3.50593e28	−0.637207
$547$	−3.00896e28	−0.536476	−0.268238	−	0.963353i	$-0.586441\pi$
$547$	−3.00896e28	−0.536476	−0.268238	+	0.963353i	$0.586441\pi$
$548$	−2.31056e28	−0.404130
$549$	−1.31411e28	−0.225487
$550$	0	0
$551$	−1.15941e29	−1.91490
$552$	−8.68717e27	−0.140773
$553$	2.07495e28	0.329909
$554$	5.70754e28	0.890424
$555$	0	0
$556$	1.04576e28	0.157090
$557$	−1.17263e29	−1.72855	−0.864277	−	0.503017i	$-0.832223\pi$
$557$	−1.17263e29	−1.72855	−0.864277	+	0.503017i	$0.832223\pi$
$558$	3.70969e28	0.536635
$559$	9.21233e28	1.30781
$560$	0	0
$561$	−2.94954e28	−0.403315
$562$	−7.71427e28	−1.03529
$563$	−6.46986e28	−0.852230	−0.426115	−	0.904669i	$-0.640118\pi$
$563$	−6.46986e28	−0.852230	−0.426115	+	0.904669i	$0.640118\pi$
$564$	−1.21289e28	−0.156816
$565$	0	0
$566$	−1.32575e29	−1.65153
$567$	−7.68456e27	−0.0939717
$568$	−3.28459e28	−0.394297
$569$	−1.35009e29	−1.59105	−0.795527	−	0.605918i	$-0.792806\pi$
$569$	−1.35009e29	−1.59105	−0.795527	+	0.605918i	$0.792806\pi$
$570$	0	0
$571$	2.55750e28	0.290494	0.145247	−	0.989395i	$-0.453602\pi$
$571$	2.55750e28	0.290494	0.145247	+	0.989395i	$0.453602\pi$
$572$	2.11515e28	0.235875
$573$	−4.82951e28	−0.528786
$574$	−1.16779e29	−1.25542
$575$	0	0
$576$	3.57664e28	0.370714
$577$	1.52580e29	1.55293	0.776465	−	0.630160i	$-0.217011\pi$
$577$	1.52580e29	1.55293	0.776465	+	0.630160i	$0.217011\pi$
$578$	9.88284e26	0.00987733
$579$	−1.06326e28	−0.104355
$580$	0	0
$581$	1.92935e26	0.00182625
$582$	−4.44702e28	−0.413407
$583$	2.85058e28	0.260264
$584$	−5.66084e28	−0.507629
$585$	0	0
$586$	−7.94506e28	−0.687341
$587$	1.08485e29	0.921868	0.460934	−	0.887434i	$-0.347514\pi$
$587$	1.08485e29	0.921868	0.460934	+	0.887434i	$0.347514\pi$
$588$	−4.50063e27	−0.0375674
$589$	3.50525e29	2.87414
$590$	0	0
$591$	9.75587e28	0.771965
$592$	−5.62023e28	−0.436894
$593$	−8.82363e28	−0.673865	−0.336933	−	0.941529i	$-0.609389\pi$
$593$	−8.82363e28	−0.673865	−0.336933	+	0.941529i	$0.609389\pi$
$594$	−1.56498e28	−0.117422
$595$	0	0
$596$	−2.32040e28	−0.168065
$597$	9.00714e28	0.640998
$598$	4.21694e28	0.294873
$599$	9.23342e28	0.634426	0.317213	−	0.948354i	$-0.397253\pi$
$599$	9.23342e28	0.634426	0.317213	+	0.948354i	$0.397253\pi$
$600$	0	0
$601$	−7.30699e28	−0.484793	−0.242397	−	0.970177i	$-0.577934\pi$
$601$	−7.30699e28	−0.484793	−0.242397	+	0.970177i	$0.577934\pi$
$602$	1.00291e29	0.653878
$603$	9.82566e27	0.0629550
$604$	5.89267e28	0.371043
$605$	0	0
$606$	1.24851e29	0.759329
$607$	−8.07544e28	−0.482709	−0.241354	−	0.970437i	$-0.577592\pi$
$607$	−8.07544e28	−0.482709	−0.241354	+	0.970437i	$0.577592\pi$
$608$	1.19396e29	0.701461
$609$	−1.03262e29	−0.596294
$610$	0	0
$611$	3.16496e29	1.76577
$612$	−1.39688e28	−0.0766070
$613$	2.21326e29	1.19316	0.596578	−	0.802555i	$-0.296527\pi$
$613$	2.21326e29	1.19316	0.596578	+	0.802555i	$0.296527\pi$
$614$	−9.08699e28	−0.481562
$615$	0	0
$616$	1.23783e29	0.633960
$617$	2.16026e29	1.08771	0.543854	−	0.839180i	$-0.316965\pi$
$617$	2.16026e29	1.08771	0.543854	+	0.839180i	$0.316965\pi$
$618$	8.85659e28	0.438417
$619$	−2.09908e29	−1.02159	−0.510795	−	0.859702i	$-0.670649\pi$
$619$	−2.09908e29	−1.02159	−0.510795	+	0.859702i	$0.670649\pi$
$620$	0	0
$621$	9.24299e27	0.0434861
$622$	−7.65609e28	−0.354167
$623$	−1.48280e29	−0.674465
$624$	−1.37940e29	−0.616953
$625$	0	0
$626$	2.15023e29	0.929938
$627$	−1.47873e29	−0.628899
$628$	2.82694e28	0.118234
$629$	1.48514e29	0.610853
$630$	0	0
$631$	4.81094e28	0.191391	0.0956955	−	0.995411i	$-0.469492\pi$
$631$	4.81094e28	0.191391	0.0956955	+	0.995411i	$0.469492\pi$
$632$	1.07580e29	0.420921
$633$	−8.01576e28	−0.308465
$634$	7.27166e28	0.275230
$635$	0	0
$636$	1.35001e28	0.0494354
$637$	1.17441e29	0.423014
$638$	−2.10296e29	−0.745100
$639$	3.49474e28	0.121802
$640$	0	0
$641$	−5.04867e28	−0.170282	−0.0851408	−	0.996369i	$-0.527134\pi$
$641$	−5.04867e28	−0.170282	−0.0851408	+	0.996369i	$0.527134\pi$
$642$	−6.52806e28	−0.216604
$643$	3.93930e29	1.28589	0.642945	−	0.765912i	$-0.277712\pi$
$643$	3.93930e29	1.28589	0.642945	+	0.765912i	$0.277712\pi$
$644$	−1.35999e28	−0.0436751
$645$	0	0
$646$	4.45545e29	1.38500
$647$	1.38475e29	0.423523	0.211762	−	0.977321i	$-0.432080\pi$
$647$	1.38475e29	0.423523	0.211762	+	0.977321i	$0.432080\pi$
$648$	−3.98420e28	−0.119896
$649$	9.67842e28	0.286573
$650$	0	0
$651$	3.12193e29	0.894998
$652$	−1.52528e28	−0.0430280
$653$	−1.16655e29	−0.323830	−0.161915	−	0.986805i	$-0.551767\pi$
$653$	−1.16655e29	−0.323830	−0.161915	+	0.986805i	$0.551767\pi$
$654$	−3.09459e28	−0.0845350
$655$	0	0
$656$	−4.59464e29	−1.21552
$657$	6.02303e28	0.156812
$658$	3.44555e29	0.882849
$659$	−3.36819e29	−0.849377	−0.424688	−	0.905340i	$-0.639616\pi$
$659$	−3.36819e29	−0.849377	−0.424688	+	0.905340i	$0.639616\pi$
$660$	0	0
$661$	4.31449e29	1.05394	0.526968	−	0.849885i	$-0.323329\pi$
$661$	4.31449e29	1.05394	0.526968	+	0.849885i	$0.323329\pi$
$662$	8.49612e28	0.204273
$663$	3.64506e29	0.862606
$664$	1.00030e27	0.00233006
$665$	0	0
$666$	7.87995e28	0.177846
$667$	1.24204e29	0.275940
$668$	−1.73894e28	−0.0380305
$669$	2.79415e29	0.601555
$670$	0	0
$671$	−2.25219e29	−0.469915
$672$	1.06340e29	0.218433
$673$	1.19710e29	0.242088	0.121044	−	0.992647i	$-0.461376\pi$
$673$	1.19710e29	0.242088	0.121044	+	0.992647i	$0.461376\pi$
$674$	−4.52008e29	−0.899948
$675$	0	0
$676$	−1.42978e29	−0.275949
$677$	−7.71503e29	−1.46608	−0.733038	−	0.680188i	$-0.761898\pi$
$677$	−7.71503e29	−1.46608	−0.733038	+	0.680188i	$0.761898\pi$
$678$	1.84722e29	0.345626
$679$	−3.74243e29	−0.689479
$680$	0	0
$681$	4.72525e29	0.844074
$682$	6.35788e29	1.11835
$683$	−6.40488e29	−1.10941	−0.554707	−	0.832046i	$-0.687170\pi$
$683$	−6.40488e29	−1.10941	−0.554707	+	0.832046i	$0.687170\pi$
$684$	−7.00316e28	−0.119455
$685$	0	0
$686$	5.76908e29	0.954340
$687$	−1.60679e29	−0.261766
$688$	3.94591e29	0.633094
$689$	−3.52277e29	−0.556650
$690$	0	0
$691$	1.07309e30	1.64482	0.822408	−	0.568898i	$-0.192630\pi$
$691$	1.07309e30	1.64482	0.822408	+	0.568898i	$0.192630\pi$
$692$	1.54094e29	0.232633
$693$	−1.31702e29	−0.195837
$694$	5.01454e28	0.0734440
$695$	0	0
$696$	−5.35382e29	−0.760794
$697$	1.21413e30	1.69950
$698$	−7.43202e29	−1.02477
$699$	−3.89791e29	−0.529446
$700$	0	0
$701$	−7.18115e29	−0.946574	−0.473287	−	0.880908i	$-0.656933\pi$
$701$	−7.18115e29	−0.946574	−0.473287	+	0.880908i	$0.656933\pi$
$702$	1.93401e29	0.251142
$703$	7.44568e29	0.952518
$704$	6.12985e29	0.772568
$705$	0	0
$706$	−6.30969e29	−0.771895
$707$	1.05070e30	1.26641
$708$	4.58362e28	0.0544327
$709$	4.24360e29	0.496534	0.248267	−	0.968692i	$-0.420139\pi$
$709$	4.24360e29	0.496534	0.248267	+	0.968692i	$0.420139\pi$
$710$	0	0
$711$	−1.14463e29	−0.130027
$712$	−7.68786e29	−0.860529
$713$	−3.75505e29	−0.414167
$714$	3.96822e29	0.431285
$715$	0	0
$716$	8.64710e28	0.0912607
$717$	−5.85898e29	−0.609356
$718$	−4.26277e29	−0.436904
$719$	−5.73272e29	−0.579038	−0.289519	−	0.957172i	$-0.593495\pi$
$719$	−5.73272e29	−0.579038	−0.289519	+	0.957172i	$0.593495\pi$
$720$	0	0
$721$	7.45334e29	0.731191
$722$	1.32527e30	1.28134
$723$	9.10400e29	0.867521
$724$	3.05133e27	0.00286572
$725$	0	0
$726$	2.87600e29	0.262395
$727$	−1.42987e30	−1.28583	−0.642916	−	0.765937i	$-0.722275\pi$
$727$	−1.42987e30	−1.28583	−0.642916	+	0.765937i	$0.722275\pi$
$728$	−1.52971e30	−1.35591
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−1.04270e30	−0.885175
$732$	−1.06662e29	−0.0892572
$733$	−1.99143e30	−1.64276	−0.821380	−	0.570381i	$-0.806795\pi$
$733$	−1.99143e30	−1.64276	−0.821380	+	0.570381i	$0.806795\pi$
$734$	−1.23405e29	−0.100351
$735$	0	0
$736$	−1.27905e29	−0.101081
$737$	1.68398e29	0.131198
$738$	6.44200e29	0.494799
$739$	−2.11124e30	−1.59872	−0.799359	−	0.600854i	$-0.794827\pi$
$739$	−2.11124e30	−1.59872	−0.799359	+	0.600854i	$0.794827\pi$
$740$	0	0
$741$	1.82743e30	1.34508
$742$	−3.83509e29	−0.278313
$743$	−1.11538e30	−0.798067	−0.399033	−	0.916936i	$-0.630654\pi$
$743$	−1.11538e30	−0.798067	−0.399033	+	0.916936i	$0.630654\pi$
$744$	1.61862e30	1.14190
$745$	0	0
$746$	−1.31331e30	−0.900763
$747$	−1.06431e27	−0.000719781 0
$748$	−2.39405e29	−0.159649
$749$	−5.49375e29	−0.361251
$750$	0	0
$751$	−2.28371e30	−1.46023	−0.730116	−	0.683323i	$-0.760534\pi$
$751$	−2.28371e30	−1.46023	−0.730116	+	0.683323i	$0.760534\pi$
$752$	1.35564e30	0.854787
$753$	5.90546e29	0.367203
$754$	2.59886e30	1.59361
$755$	0	0
$756$	−6.23732e28	−0.0371980
$757$	−9.73411e29	−0.572519	−0.286259	−	0.958152i	$-0.592412\pi$
$757$	−9.73411e29	−0.572519	−0.286259	+	0.958152i	$0.592412\pi$
$758$	2.80951e29	0.162969
$759$	1.58412e29	0.0906251
$760$	0	0
$761$	−3.06985e30	−1.70836	−0.854179	−	0.519979i	$-0.825940\pi$
$761$	−3.06985e30	−1.70836	−0.854179	+	0.519979i	$0.825940\pi$
$762$	4.15933e29	0.228295
$763$	−2.60428e29	−0.140987
$764$	−3.91996e29	−0.209316
$765$	0	0
$766$	2.02936e30	1.05428
$767$	−1.19607e30	−0.612920
$768$	7.39078e29	0.373592
$769$	−7.51816e29	−0.374874	−0.187437	−	0.982277i	$-0.560018\pi$
$769$	−7.51816e29	−0.374874	−0.187437	+	0.982277i	$0.560018\pi$
$770$	0	0
$771$	−1.81042e30	−0.878432
$772$	−8.63013e28	−0.0413081
$773$	−1.23771e30	−0.584434	−0.292217	−	0.956352i	$-0.594393\pi$
$773$	−1.23771e30	−0.584434	−0.292217	+	0.956352i	$0.594393\pi$
$774$	−5.53243e29	−0.257713
$775$	0	0
$776$	−1.94033e30	−0.879686
$777$	6.63144e29	0.296611
$778$	−4.58507e29	−0.202330
$779$	6.08698e30	2.65008
$780$	0	0
$781$	5.98949e29	0.253836
$782$	−4.77297e29	−0.199580
$783$	5.69637e29	0.235017
$784$	5.03033e29	0.204776
$785$	0	0
$786$	1.91247e30	0.757979
$787$	−2.14352e30	−0.838288	−0.419144	−	0.907920i	$-0.637670\pi$
$787$	−2.14352e30	−0.838288	−0.419144	+	0.907920i	$0.637670\pi$
$788$	7.91853e29	0.305576
$789$	−1.37853e30	−0.524936
$790$	0	0
$791$	1.55454e30	0.576434
$792$	−6.82834e29	−0.249862
$793$	2.78327e30	1.00505
$794$	1.87221e30	0.667174
$795$	0	0
$796$	7.31081e29	0.253734
$797$	−3.01280e30	−1.03195	−0.515974	−	0.856605i	$-0.672570\pi$
$797$	−3.01280e30	−1.03195	−0.515974	+	0.856605i	$0.672570\pi$
$798$	1.98944e30	0.672513
$799$	−3.58228e30	−1.19514
$800$	0	0
$801$	8.17975e29	0.265827
$802$	1.27705e30	0.409617
$803$	1.03226e30	0.326796
$804$	7.97518e28	0.0249202
$805$	0	0
$806$	−7.85711e30	−2.39191
$807$	−1.35657e30	−0.407634
$808$	5.44752e30	1.61577
$809$	−5.76883e30	−1.68900	−0.844498	−	0.535559i	$-0.820101\pi$
$809$	−5.76883e30	−1.68900	−0.844498	+	0.535559i	$0.820101\pi$
$810$	0	0
$811$	3.98728e30	1.13752	0.568759	−	0.822504i	$-0.307424\pi$
$811$	3.98728e30	1.13752	0.568759	+	0.822504i	$0.307424\pi$
$812$	−8.38148e29	−0.236038
$813$	2.37427e30	0.660055
$814$	1.35051e30	0.370631
$815$	0	0
$816$	1.56129e30	0.417576
$817$	−5.22754e30	−1.38027
$818$	−5.61998e30	−1.46496
$819$	1.62759e30	0.418855
$820$	0	0
$821$	−4.09615e30	−1.02748	−0.513739	−	0.857946i	$-0.671740\pi$
$821$	−4.09615e30	−1.02748	−0.513739	+	0.857946i	$0.671740\pi$
$822$	−3.62085e30	−0.896719
$823$	−1.43384e30	−0.350592	−0.175296	−	0.984516i	$-0.556088\pi$
$823$	−1.43384e30	−0.350592	−0.175296	+	0.984516i	$0.556088\pi$
$824$	3.86432e30	0.932905
$825$	0	0
$826$	−1.30210e30	−0.306447
$827$	3.13142e30	0.727668	0.363834	−	0.931464i	$-0.381468\pi$
$827$	3.13142e30	0.727668	0.363834	+	0.931464i	$0.381468\pi$
$828$	7.50225e28	0.0172136
$829$	−4.44329e30	−1.00666	−0.503329	−	0.864095i	$-0.667892\pi$
$829$	−4.44329e30	−1.00666	−0.503329	+	0.864095i	$0.667892\pi$
$830$	0	0
$831$	−2.64966e30	−0.585301
$832$	−7.57531e30	−1.65236
$833$	−1.32926e30	−0.286311
$834$	1.63880e30	0.348564
$835$	0	0
$836$	−1.20024e30	−0.248944
$837$	−1.72218e30	−0.352745
$838$	−3.00994e30	−0.608829
$839$	−9.30057e29	−0.185784	−0.0928922	−	0.995676i	$-0.529611\pi$
$839$	−9.30057e29	−0.185784	−0.0928922	+	0.995676i	$0.529611\pi$
$840$	0	0
$841$	2.52173e30	0.491293
$842$	3.46057e30	0.665842
$843$	3.58126e30	0.680528
$844$	−6.50614e29	−0.122103
$845$	0	0
$846$	−1.90070e30	−0.347957
$847$	2.42033e30	0.437621
$848$	−1.50891e30	−0.269467
$849$	6.15462e30	1.08560
$850$	0	0
$851$	−7.97630e29	−0.137259
$852$	2.83657e29	0.0482145
$853$	2.27471e30	0.381911	0.190955	−	0.981599i	$-0.438841\pi$
$853$	2.27471e30	0.381911	0.190955	+	0.981599i	$0.438841\pi$
$854$	3.03003e30	0.502503
$855$	0	0
$856$	−2.84833e30	−0.460910
$857$	−3.05003e30	−0.487534	−0.243767	−	0.969834i	$-0.578383\pi$
$857$	−3.05003e30	−0.487534	−0.243767	+	0.969834i	$0.578383\pi$
$858$	3.31462e30	0.523380
$859$	1.34073e30	0.209128	0.104564	−	0.994518i	$-0.466655\pi$
$859$	1.34073e30	0.209128	0.104564	+	0.994518i	$0.466655\pi$
$860$	0	0
$861$	5.42133e30	0.825225
$862$	1.12873e31	1.69732
$863$	−1.14916e31	−1.70713	−0.853566	−	0.520985i	$-0.825565\pi$
$863$	−1.14916e31	−1.70713	−0.853566	+	0.520985i	$0.825565\pi$
$864$	−5.86612e29	−0.0860908
$865$	0	0
$866$	4.32457e30	0.619448
$867$	−4.58799e28	−0.00649265
$868$	2.53397e30	0.354278
$869$	−1.96173e30	−0.270976
$870$	0	0
$871$	−2.08107e30	−0.280606
$872$	−1.35024e30	−0.179882
$873$	2.06448e30	0.271744
$874$	−2.39290e30	−0.311211
$875$	0	0
$876$	4.88871e29	0.0620727
$877$	−9.06320e30	−1.13707	−0.568533	−	0.822660i	$-0.692489\pi$
$877$	−9.06320e30	−1.13707	−0.568533	+	0.822660i	$0.692489\pi$
$878$	2.91934e30	0.361904
$879$	3.68840e30	0.451809
$880$	0	0
$881$	−6.50767e30	−0.778356	−0.389178	−	0.921163i	$-0.627241\pi$
$881$	−6.50767e30	−0.778356	−0.389178	+	0.921163i	$0.627241\pi$
$882$	−7.05287e29	−0.0833577
$883$	−5.90742e30	−0.689939	−0.344969	−	0.938614i	$-0.612111\pi$
$883$	−5.90742e30	−0.689939	−0.344969	+	0.938614i	$0.612111\pi$
$884$	2.95858e30	0.341456
$885$	0	0
$886$	7.18992e30	0.810344
$887$	3.42312e30	0.381263	0.190632	−	0.981662i	$-0.438946\pi$
$887$	3.42312e30	0.381263	0.190632	+	0.981662i	$0.438946\pi$
$888$	3.43819e30	0.378437
$889$	3.50032e30	0.380750
$890$	0	0
$891$	7.26524e29	0.0771852
$892$	2.26792e30	0.238121
$893$	−1.79595e31	−1.86361
$894$	−3.63627e30	−0.372917
$895$	0	0
$896$	−4.47020e30	−0.447810
$897$	−1.95766e30	−0.193828
$898$	1.56863e31	1.53503
$899$	−2.31420e31	−2.23833
$900$	0	0
$901$	3.98728e30	0.376761
$902$	1.10407e31	1.03116
$903$	−4.65587e30	−0.429813
$904$	8.05980e30	0.735455
$905$	0	0
$906$	9.23433e30	0.823302
$907$	1.16251e31	1.02452	0.512262	−	0.858829i	$-0.328808\pi$
$907$	1.16251e31	1.02452	0.512262	+	0.858829i	$0.328808\pi$
$908$	3.83534e30	0.334120
$909$	−5.79606e30	−0.499128
$910$	0	0
$911$	−1.34510e31	−1.13191	−0.565957	−	0.824435i	$-0.691493\pi$
$911$	−1.34510e31	−1.13191	−0.565957	+	0.824435i	$0.691493\pi$
$912$	7.82741e30	0.651137
$913$	−1.82407e28	−0.00150002
$914$	9.70051e30	0.788605
$915$	0	0
$916$	−1.30418e30	−0.103618
$917$	1.60945e31	1.26416
$918$	−2.18903e30	−0.169982
$919$	−8.55562e30	−0.656808	−0.328404	−	0.944537i	$-0.606511\pi$
$919$	−8.55562e30	−0.656808	−0.328404	+	0.944537i	$0.606511\pi$
$920$	0	0
$921$	4.21853e30	0.316544
$922$	−1.15093e30	−0.0853835
$923$	−7.40185e30	−0.542903
$924$	−1.06899e30	−0.0775205
$925$	0	0
$926$	1.52259e30	0.107937
$927$	−4.11157e30	−0.288184
$928$	−7.88268e30	−0.546286
$929$	9.93712e30	0.680919	0.340460	−	0.940259i	$-0.389417\pi$
$929$	9.93712e30	0.680919	0.340460	+	0.940259i	$0.389417\pi$
$930$	0	0
$931$	−6.66418e30	−0.446452
$932$	−3.16381e30	−0.209577
$933$	3.55425e30	0.232804
$934$	1.48757e31	0.963462
$935$	0	0
$936$	8.43852e30	0.534404
$937$	2.14272e30	0.134183	0.0670917	−	0.997747i	$-0.478628\pi$
$937$	2.14272e30	0.134183	0.0670917	+	0.997747i	$0.478628\pi$
$938$	−2.26557e30	−0.140297
$939$	−9.98217e30	−0.611275
$940$	0	0
$941$	1.69148e31	1.01292	0.506459	−	0.862264i	$-0.330954\pi$
$941$	1.69148e31	1.01292	0.506459	+	0.862264i	$0.330954\pi$
$942$	4.43006e30	0.262347
$943$	−6.52077e30	−0.381879
$944$	−5.12310e30	−0.296706
$945$	0	0
$946$	−9.48180e30	−0.537074
$947$	1.96530e31	1.10092	0.550459	−	0.834862i	$-0.314453\pi$
$947$	1.96530e31	1.10092	0.550459	+	0.834862i	$0.314453\pi$
$948$	−9.29057e29	−0.0514701
$949$	−1.27568e31	−0.698948
$950$	0	0
$951$	−3.37578e30	−0.180917
$952$	1.73142e31	0.917728
$953$	−2.39001e31	−1.25292	−0.626462	−	0.779452i	$-0.715498\pi$
$953$	−2.39001e31	−1.25292	−0.626462	+	0.779452i	$0.715498\pi$
$954$	2.11559e30	0.109692
$955$	0	0
$956$	−4.75555e30	−0.241209
$957$	9.76276e30	0.489776
$958$	2.60567e31	1.29295
$959$	−3.04716e31	−1.49555
$960$	0	0
$961$	4.91397e31	2.35959
$962$	−1.66897e31	−0.792703
$963$	3.03057e30	0.142380
$964$	7.38943e30	0.343401
$965$	0	0
$966$	−2.13122e30	−0.0969100
$967$	−1.63072e31	−0.733504	−0.366752	−	0.930319i	$-0.619530\pi$
$967$	−1.63072e31	−0.733504	−0.366752	+	0.930319i	$0.619530\pi$
$968$	1.25486e31	0.558348
$969$	−2.06839e31	−0.910401
$970$	0	0
$971$	3.09343e31	1.33241	0.666207	−	0.745767i	$-0.267917\pi$
$971$	3.09343e31	1.33241	0.666207	+	0.745767i	$0.267917\pi$
$972$	3.44076e29	0.0146608
$973$	1.37915e31	0.581334
$974$	4.77865e30	0.199267
$975$	0	0
$976$	1.19216e31	0.486531
$977$	−1.98044e31	−0.799592	−0.399796	−	0.916604i	$-0.630919\pi$
$977$	−1.98044e31	−0.799592	−0.399796	+	0.916604i	$0.630919\pi$
$978$	−2.39025e30	−0.0954742
$979$	1.40189e31	0.553982
$980$	0	0
$981$	1.43663e30	0.0555673
$982$	−1.94829e31	−0.745561
$983$	4.34284e30	0.164423	0.0822114	−	0.996615i	$-0.473802\pi$
$983$	4.34284e30	0.164423	0.0822114	+	0.996615i	$0.473802\pi$
$984$	2.81078e31	1.05288
$985$	0	0
$986$	−2.94154e31	−1.07862
$987$	−1.59955e31	−0.580322
$988$	1.48327e31	0.532441
$989$	5.60008e30	0.198899
$990$	0	0
$991$	4.70795e30	0.163704	0.0818518	−	0.996645i	$-0.473917\pi$
$991$	4.70795e30	0.163704	0.0818518	+	0.996645i	$0.473917\pi$
$992$	2.38317e31	0.819939
$993$	−3.94422e30	−0.134275
$994$	−8.05807e30	−0.271440
$995$	0	0
$996$	−8.63864e27	−0.000284920 0
$997$	1.58934e31	0.518700	0.259350	−	0.965783i	$-0.416492\pi$
$997$	1.58934e31	0.518700	0.259350	+	0.965783i	$0.416492\pi$
$998$	4.89494e30	0.158080
$999$	−3.65817e30	−0.116903

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.d.1.2		2
5.2	odd	4		75.22.b.d.49.3		4
5.3	odd	4		75.22.b.d.49.2		4
5.4	even	2		3.22.a.c.1.1	✓	2
15.14	odd	2		9.22.a.e.1.2		2
20.19	odd	2		48.22.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.22.a.c.1.1	✓	2	5.4	even	2
9.22.a.e.1.2		2	15.14	odd	2
48.22.a.g.1.1		2	20.19	odd	2
75.22.a.d.1.2		2	1.1	even	1	trivial
75.22.b.d.49.2		4	5.3	odd	4
75.22.b.d.49.3		4	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+1271.96 q^{2} -59049.0 q^{3} -479282. q^{4} -7.51077e7 q^{6} -6.32076e8 q^{7} -3.27711e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q+1271.96 q^{2} -59049.0 q^{3} -479282. q^{4} -7.51077e7 q^{6} -6.32076e8 q^{7} -3.27711e9 q^{8} +3.48678e9 q^{9} +5.97585e10 q^{11} +2.83011e10 q^{12} -7.38499e11 q^{13} -8.03972e11 q^{14} -3.16321e12 q^{16} +8.35876e12 q^{17} +4.43503e12 q^{18} +4.19061e13 q^{19} +3.73234e13 q^{21} +7.60101e13 q^{22} -4.48926e13 q^{23} +1.93510e14 q^{24} -9.39338e14 q^{26} -2.05891e14 q^{27} +3.02943e14 q^{28} -2.76669e15 q^{29} +8.36452e15 q^{31} +2.84914e15 q^{32} -3.52868e15 q^{33} +1.06320e16 q^{34} -1.67115e15 q^{36} +1.77675e16 q^{37} +5.33027e16 q^{38} +4.36077e16 q^{39} +1.45253e17 q^{41} +4.74737e16 q^{42} -1.24744e17 q^{43} -2.86412e16 q^{44} -5.71014e16 q^{46} -4.28566e17 q^{47} +1.86784e17 q^{48} -1.59026e17 q^{49} -4.93577e17 q^{51} +3.53950e17 q^{52} +4.77017e17 q^{53} -2.61884e17 q^{54} +2.07138e18 q^{56} -2.47452e18 q^{57} -3.51911e18 q^{58} +1.61959e18 q^{59} -3.76882e18 q^{61} +1.06393e19 q^{62} -2.20391e18 q^{63} +1.02577e19 q^{64} -4.48832e18 q^{66} +2.81797e18 q^{67} -4.00621e18 q^{68} +2.65086e18 q^{69} +1.00228e19 q^{71} -1.14266e19 q^{72} +1.72739e19 q^{73} +2.25995e19 q^{74} -2.00849e19 q^{76} -3.77719e19 q^{77} +5.54670e19 q^{78} -3.28276e19 q^{79} +1.21577e19 q^{81} +1.84755e20 q^{82} -3.05240e17 q^{83} -1.78885e19 q^{84} -1.58669e20 q^{86} +1.63370e20 q^{87} -1.95835e20 q^{88} +2.34593e20 q^{89} +4.66787e20 q^{91} +2.15162e19 q^{92} -4.93917e20 q^{93} -5.45116e20 q^{94} -1.68239e20 q^{96} +5.92086e20 q^{97} -2.02274e20 q^{98} +2.08365e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 666 q^{2} - 118098 q^{3} + 1179236 q^{4} + 39326634 q^{6} - 679896112 q^{7} - 2427055848 q^{8} + 6973568802 q^{9}+O(q^{10}) \) 2 * q - 666 * q^2 - 118098 * q^3 + 1179236 * q^4 + 39326634 * q^6 - 679896112 * q^7 - 2427055848 * q^8 + 6973568802 * q^9	\( 2 q - 666 q^{2} - 118098 q^{3} + 1179236 q^{4} + 39326634 q^{6} - 679896112 q^{7} - 2427055848 q^{8} + 6973568802 q^{9} + 219869122968 q^{11} - 69632706564 q^{12} + 48468909956 q^{13} - 711297706896 q^{14} - 8288736440560 q^{16} + 11333529041436 q^{17} - 2322198411066 q^{18} + 11960585011624 q^{19} + 40147185517488 q^{21} - 234277148563128 q^{22} + 146508390063504 q^{23} + 143315220768552 q^{24} - 24\!\cdots\!16 q^{26}+ \cdots + 76\!\cdots\!68 q^{99}+O(q^{100}) \) 2 * q - 666 * q^2 - 118098 * q^3 + 1179236 * q^4 + 39326634 * q^6 - 679896112 * q^7 - 2427055848 * q^8 + 6973568802 * q^9 + 219869122968 * q^11 - 69632706564 * q^12 + 48468909956 * q^13 - 711297706896 * q^14 - 8288736440560 * q^16 + 11333529041436 * q^17 - 2322198411066 * q^18 + 11960585011624 * q^19 + 40147185517488 * q^21 - 234277148563128 * q^22 + 146508390063504 * q^23 + 143315220768552 * q^24 - 2464447363969716 * q^26 - 411782264189298 * q^27 + 223631286954656 * q^28 - 1798520043674052 * q^29 + 11169107526944992 * q^31 + 10999491515934048 * q^32 - 12983051842137432 * q^33 + 4867007487413652 * q^34 + 4111741689897636 * q^36 - 12736264858660012 * q^37 + 111335877667820664 * q^38 - 2862040663991844 * q^39 + 122972020616468052 * q^41 + 42001418294501904 * q^42 - 288455418162270040 * q^43 + 236905239135592368 * q^44 - 428027981395161168 * q^46 - 837243745741596960 * q^47 + 489441598078627440 * q^48 - 715285396941470670 * q^49 - 669233556367754364 * q^51 + 1659150748242045896 * q^52 + 43007964012775764 * q^53 + 137123493975036234 * q^54 + 2030730320050568640 * q^56 - 706260584351385576 * q^57 - 5395375030104291852 * q^58 - 3523823330903857224 * q^59 - 1779023128451013860 * q^61 + 5204128043677477536 * q^62 - 2370651157622148912 * q^63 + 5211692013463482944 * q^64 + 13833831345504145272 * q^66 + 16454068667621610296 * q^67 + 927498335984326008 * q^68 - 8651173924859847696 * q^69 + 17379227131150420944 * q^71 - 8462620471162227048 * q^72 - 50891146268473989076 * q^73 + 81714426393675287484 * q^74 - 69750112936092790064 * q^76 - 45428450983816025664 * q^77 + 145523152395047760084 * q^78 - 54055785594190591040 * q^79 + 24315330918113857602 * q^81 + 227933719179803895996 * q^82 - 111108429277666677288 * q^83 - 13205203863385482144 * q^84 + 158596867265246585208 * q^86 + 106200810058909096548 * q^87 - 59732519594715541728 * q^88 + 226920767965448065524 * q^89 + 429154174685185000352 * q^91 + 338958281810136649248 * q^92 - 659524630358574832608 * q^93 + 246883115968835119776 * q^94 - 649508974524389600352 * q^96 + 128331469252795746236 * q^97 + 875730737371150330614 * q^98 + 766636228226373222168 * q^99

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